Resolution of Algebraical Equations. 137 



Hence a^ + a„ = coefficient of- in — 1. ^-V*. 



x x' 



Now since t^ is a root, therefore (x) is of the form #-<*, . P; 



therefore a, + a„ = coefficient of - in - 1. 1 - — - 1. — 



X XX 



1 . p 



= a, — coefficient of - in 1. — , 



1 P 



but the coefficient of - in -1.— is the least root of the equation 



P=0 (by Section 1). 



Hence a„ must be the least of the quantities: «„, a 3 , « 4 , &c. : by 

 similar proof a, must be the least of a 13 a 3 , a 4 , &c. ; and therefore 

 «, and «„ are the two least of a lt a„, a 3 , &c, that is, they are the 

 two least roots. 



In the. same manner if by the present method we get the sum 

 of three roots a,+a„ + a 3 , then putting <p(x) = x — a l .x — a 3 .P', we 

 have 



_L*J — l(i-&)-L(i-SWl£. 



x' \ X' \ xl X 



and taking the coefficients of - at both sides by the theorems 



already established, we have 



a, + a„ + a 3 = a 1 + a„ + least root of the equation P=0; 

 therefore a a is the least of the quantities a 3 , a 4 , a 5 , &c. 

 Similarly a 2 o 2 , a 4 , a 5 , &c. 



«i a,, a 4 , an, &C. 



therefore a,, a„, a 3 , are the 3 least roots of the equation: thus it 

 appears in general that this method gives the sum of the m least 

 roots. 



Vol. IV. Part I. S 



